Significant Figures Rules and Uncertainty 3

Significant figures express the uncertainty or precision of a measurement. Every measurement carries some inherent uncertainty, whether due to the instrument used or human limitations. Scientists use significant figures to communicate the reliability of measured and calculated values in a standardized way.

For instance, imagine you’re in a chemistry lab measuring 8 mL of liquid. If you pour it by eye into a beaker without markings, your estimate may vary widely. If the beaker has 5 mL divisions, you might be accurate within ±2 mL. But with a graduated cylinder marked to the nearest 0.1 mL, you can report the value as 8.0 mL, accurate to ±0.1 mL. The number of digits you record reflects the resolution of the tool. This is where significant figures come in.

Key Takeaways: Significant Figures Rules

• Significant figures (or significant digits) indicate the precision of a measured or calculated value.
• More significant figures reflect a higher level of measurement precision.
• Zeros may or may not be significant, depending on their position in the number.
• Scientific notation helps clearly identify the number of significant figures.
• Addition/subtraction uses decimal precision; multiplication/division uses the number of significant figures.
• Exact numbers (like counts or defined constants) do not limit significant figures.
• Significant figures can be lost in subtraction or rounding, even if starting with high-precision measurements.

Free Significant Figures Quick Reference

Need a handy summary of the rules for significant figures? Download our quick reference chart, available as both a PDF for printing and a PNG for use in presentations or digital lessons. This resource is designed for students and teachers who want a clear, at-a-glance guide to significant figure rules and examples.

What Are Significant Figures?

Significant figures are the digits in a number that convey meaningful information about its precision. These include:

1. All non-zero digits.
2. Any zeros between non-zero digits.
3. Trailing zeros in a decimal number.
4. Leading zeros are not significant. They serve only to position the decimal point.
5. Exact numbers have an infinite number of significant figures.

Significant Figure Rules

Significant figures express uncertainty or precision. The more significant figures in a measurement, the more precise the measurement. There are six basic rules dealing with significant figures.

Here are six basic rules:

1. Non-zero digits are always significant.
   • Example: 345 has three significant figures.
2. Zeros between significant digits are significant.
   • Example: 205 has three significant figures.
3. Leading zeros are not significant.
   • Example: 0.0032 has two significant figures.
4. Trailing zeros after a decimal point are significant.
   • Example: 2.00 has three significant figures.
5. Trailing zeros in a whole number without a decimal may not be significant.
   • Example: 2300 has two significant figures (unless written as 2.300 × 10³).
6. Exact numbers (e.g., counting numbers or defined conversions like 1 in = 2.54 cm) have infinite significant figures.

Most and Least Significant Figures

• Most significant figure (MSF): The first non-zero digit from the left.
• Least significant figure (LSF): The last digit (whether zero or non-zero) on the right.

Example:

• In 0.00321, the MSF is 3 and the LSF is 1.
• In 4.320, the MSF is 4 and the LSF is 0 (which is significant).

Quick Tip: Use Scientific Notation

Converting a number to scientific notation makes it easy to count significant figures.

Example: How many significant figures are in each?

Significant Figures in Calculations

Once you have measurements, using them in calculations requires rules to preserve proper precision.

Addition and Subtraction

• Round the result to the least precise decimal place.
• Ignore the number of significant figures.

Example:
24.21 cm + 5.005 cm + 22 cm = 51.215 cm → round to 51 cm (since 22 cm is least precise).

Multiplication and Division

• Round the result to the smallest number of significant figures.

Example:
35.105 g ÷ 35 mL = 1.003 g/mL → round to 1.0 g/mL (35 has 2 sig figs).

Losing Significant Figures

Significant digits can be lost during subtraction or rounding:

Example:
Beaker + water = 75.844 g
Empty beaker = 75.206 g
Water mass = 75.844 – 75.206 = 0.638 g (only 3 significant figures remain)

Even though both initial values had 5 significant figures, the subtraction reduces precision.

Exact Numbers and Constants

Some numbers are defined exactly and don’t limit the precision of a calculation. These include:

• Counting numbers (e.g., 3 apples)
• Defined quantities (e.g., 1 inch = 2.54 cm)
• Some constants (e.g., π)

Example:
To average 10.3 cm, 12.7 cm, and 14.5 cm:
(10.3 + 12.7 + 14.5) ÷ 3 = 37.5 ÷ 3 = 12.5 cm
Even though “3” has only one digit, it’s an exact number and doesn’t reduce the sig figs.

Why Significant Figures Matter

Significant figures prevent overstatement of precision. They are essential in fields like chemistry, physics, engineering, and environmental science (anywhere measurements and calculations impact decisions or results).

Learning to use them properly ensures consistency, clarity, and scientific integrity.

Significant Figures Worksheets

Practice working with significant figures using worksheets:

• Addition and subtraction worksheet [PDF Worksheet] [Answer Key]
• Multiplication and division worksheet [PDF Worksheet] [Answer Key]
• Significant figures in decimals and scientific notation [PDF Worksheet] [Answer Key]

More About Measurements

Learn more about significant figures and measurements:

• What is the Difference Between Accuracy and Precision?
• Why Use 4 Significant Figures for Avogadro’s Number?
• Significant Figures Periodic Table

Common Mistakes With Significant Figures

Even experienced students sometimes slip up when applying sig fig rules. Here are some of the most common mistakes and how to avoid them:

Treating All Zeros as Significant

Not all zeros count! Only zeros between significant digits, or trailing zeros after a decimal, are significant.

• Wrong: Thinking 0.0020 has one sig fig.
• Right: It has two sig figs (the 2 and the trailing 0).

Forgetting to Round Correctly After Calculations

When multiplying or dividing, use the least number of sig figs. When adding or subtracting, match the least number of decimal places.

• Wrong: Reporting 3.42 × 5.1 = 17.442 as-is.
• Right: Round to 2 sig figs → 17.

Misinterpreting Whole Numbers Without Decimals

Trailing zeros in a number like 1500 can be ambiguous.

• 1500 → could be 2, 3, or 4 sig figs depending on context.
• Use scientific notation to clarify:
  • 1.5 × 10³ → 2 sig figs
  • 1.500 × 10³ → 4 sig figs

Limiting Precision Based on Exact Numbers

Do not apply sig fig rules to exact counts or defined conversions.

• Wrong: Dividing a sum by “3” and rounding to 1 sig fig.
• Right: Exact numbers do not reduce the precision of your result.

Over-rounding Too Early

Round only after the final step in a multistep calculation. Rounding intermediate results can compound errors.

• Tip: Keep extra digits in intermediate steps, then round your final answer according to sig fig rules.

References and Further Reading

• Delury, D. B. (1958). “Computations with approximate numbers”. The Mathematics Teacher. 51 (7): 521–30. doi:10.5951/MT.51.7.0521
• Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM.
• Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry. Austin, Texas: Holt Rinehart Winston. ISBN 0-03-052002-9.